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34
Counting independent sets up to the tree threshold
 In STOC ’06: Proceedings of the thirtyeighth annual ACM symposium on Theory of computing
, 2006
"... Consider the problem of approximately counting weighted independent sets of a graph G with activity λ, i.e., where the weight of an independent set I is λ I . We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for any graph of maximum degree Δ ..."
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Cited by 89 (1 self)
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Consider the problem of approximately counting weighted independent sets of a graph G with activity λ, i.e., where the weight of an independent set I is λ I . We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for any graph of maximum degree Δ and λ<λc =(Δ − 1) Δ−1 /(Δ − 2) Δ.Thisimproves on the previously known general bound of λ ≤ 2
On the Relative Complexity of Approximate Counting Problems
, 2000
"... Two natural classes of counting problems that are interreducible under approximationpreserving reductions are: (i) those that admit a particular kind of ecient approximation algorithm known as an \FPRAS," and (ii) those that are complete for #P with respect to approximationpreserving reduc ..."
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Cited by 58 (22 self)
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Two natural classes of counting problems that are interreducible under approximationpreserving reductions are: (i) those that admit a particular kind of ecient approximation algorithm known as an \FPRAS," and (ii) those that are complete for #P with respect to approximationpreserving reducibility. We describe and investigate not only these two classes but also a third class, of intermediate complexity, that is not known to be identical to (i) or (ii). The third class can be characterised as the hardest problems in a logically dened subclass of #P. Research Report 370, Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK. This work was supported in part by the EPSRC Research Grant \Sharper Analysis of Randomised Algorithms: a Computational Approach" and by the ESPRIT Projects RANDAPX and ALCOMFT. y dyer@scs.leeds.ac.uk, School of Computer Studies, University of Leeds, Leeds LS2 9JT, United Kingdom. z leslie@dcs.warwick.ac.uk, http://www.dcs.warw...
The repulsive lattice gas, the independentset polynomial, and the Lovász local lemma
, 2004
"... We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {px} if and only if the independent ..."
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Cited by 45 (7 self)
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We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {px} if and only if the independentset polynomial for G is nonvanishing in the polydisc of radii {px}. Furthermore, we show that the usual proof of the Lovász local lemma — which provides a sufficient condition for this to occur — corresponds to a simple inductive argument for the nonvanishing of the independentset polynomial in a polydisc, which was discovered implicitly by Shearer [98] and explicitly by Dobrushin [37, 38]. We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for “soft” dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternatingsign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.
Markov Random Fields and Percolation on General Graphs
 Adv. Appl. Probab
, 1999
"... Let G be an infinite, locally finite, connected graph with bounded degree. We show that G supports phase transition in all or none of the following five models: bond percolation, site percolation, the Ising model, the WidomRowlinson model and the beach model. Some, but not all, of these implicatio ..."
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Cited by 26 (3 self)
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Let G be an infinite, locally finite, connected graph with bounded degree. We show that G supports phase transition in all or none of the following five models: bond percolation, site percolation, the Ising model, the WidomRowlinson model and the beach model. Some, but not all, of these implications hold without the bounded degree assumption. We finally give two examples of (random) unbounded degree graphs in which phase transition in all five models can be established: supercritical GaltonWatson trees, and PoissonVoronoi tessellations of R d for d 2. Keywords: Percolation, Ising model, WidomRowlinson model, beach model, GaltonWatson tree, PoissonVoronoi tessellation. AMS Subject Classification: Primary 60K35, Secondary 82B20, 82B43 1 Introduction Over the last few decades, it has become increasingly clear that there are important connections between percolation theory on one hand, and the issue of Gibbs state multiplicity in Markov random fields on the other. Example...
Random Colorings of a Cayley Tree
 IN CONTEMPORARY COMBINATORICS, B. BOLLOBAS, ED., BOLYAI SOCIETY MATHEMATICAL STUDIES, 2002
, 2000
"... Probability measures on the space of proper colorings of a Cayley tree (that is, an infinite regular connected graph with no cycles) are of interest not only in combinatorics but also in statistical physics, as states of the antiferromagnetic Potts model at zero temperature, on the "Bethe latti ..."
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Cited by 22 (1 self)
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Probability measures on the space of proper colorings of a Cayley tree (that is, an infinite regular connected graph with no cycles) are of interest not only in combinatorics but also in statistical physics, as states of the antiferromagnetic Potts model at zero temperature, on the "Bethe lattice". We concentrate on a particularly nice class of such measures which remain invariant under paritypreserving automorphisms of the tree. Making use of a correspondence with branching random walks on certain bipartite graphs, we determine when more than one such measure exists. The case of "uniform" measures is particularly interesting, and as it turns out, plays a special role. Some of the results herein are deducible from previous work of the authors and by members of the statistical physics community, but many are new. We hope that this work will serve as a helpful glimpse into the rapidly expanding intersection of combinatorics and statistical physics.
A personal list of unsolved problems concerning Potts models and lattice gases
 TO APPEAR IN MARKOV PROCESSES AND RELATED FIELDS
, 2000
"... I review recent results and unsolved problems concerning the hardcore lattice gas and the qcoloring model (antiferromagnetic Potts model at zero temperature). For each model, I consider its equilibrium properties (uniqueness/nonuniqueness of the infinitevolume Gibbs measure, complex zeros of the ..."
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Cited by 17 (1 self)
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I review recent results and unsolved problems concerning the hardcore lattice gas and the qcoloring model (antiferromagnetic Potts model at zero temperature). For each model, I consider its equilibrium properties (uniqueness/nonuniqueness of the infinitevolume Gibbs measure, complex zeros of the partition function) and the dynamics of local and nonlocal Monte Carlo algorithms (ergodicity, rapid mixing, mixing at complex fugacity). These problems touch on mathematical physics, probability, combinatorics and theoretical computer science.
Exact and approximate results for deposition and annihilation processes on graphs.
 Ann. Appl. Prob.
, 2005
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Improved mixing condition on the grid for counting and sampling independent sets
 In Proceedings of the 52nd Symposium on Foundations of Computer Science
, 2011
"... The hardcore model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics, a multicast model of calls in communication networks, and as a weighted independent set problem in combinatorics, probability and theoretical computer sc ..."
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Cited by 11 (3 self)
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The hardcore model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics, a multicast model of calls in communication networks, and as a weighted independent set problem in combinatorics, probability and theoretical computer science. In this model, each independent set I in a graph G is weighted proportionally to λI, for a positive real parameter λ. For large λ, computing the partition function (namely, the normalizing constant which makes the weighting a probability distribution on a finite graph) on graphs of maximum degree ∆ ≥ 3, is a well known computationally challenging problem. More concretely, let λc(T∆) denote the critical value for the socalled uniqueness threshold of the hardcore model on the infinite ∆regular tree; recent breakthrough results of Dror Weitz (2006) and Allan Sly (2010) have identified λc(T∆) as a threshold where the hardness of estimating the above partition function undergoes a computational transition. We focus on the wellstudied particular case of the square lattice Z2, and provide a new lower bound for the uniqueness threshold, in particular taking it well above λc(T4). Our
Hard Constraints and the Bethe Lattice: Adventures at the Interface of Combinatorics and Statistical Physics
, 2002
"... Statistical physics models with hard constraints, such as the discrete hardcore gas model (random independent sets in a graph), are inherently combinatorial and present the discrete mathematician with a relatively comfortable setting for the study of phase transition. In this paper we survey recent ..."
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Cited by 8 (0 self)
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Statistical physics models with hard constraints, such as the discrete hardcore gas model (random independent sets in a graph), are inherently combinatorial and present the discrete mathematician with a relatively comfortable setting for the study of phase transition. In this paper we survey recent work (concentrating on joint work of the authors) in which hardconstraint systems are modeled by the space Hom(G, H) of homomorphisms from an infinite graph G to a fixed finite constraint graph H. These spaces become sufficiently tractable when G is a regular tree (often called a Cayley tree or Bethe lattice) to permit characterization of the constraint graphs H which admit multiple invariant Gibbs measures. Applications to a physics problem (multiple critical points for symmetrybreaking) and a combinatorics problem (random coloring), as well as some new combinatorial notions, will be presented.